{
  "id": "1812.10938",
  "version": "v1",
  "published": "2018-12-28T09:52:44.000Z",
  "updated": "2018-12-28T09:52:44.000Z",
  "title": "Variations and extensions of the Gaussian concentration inequality",
  "authors": [
    "Daniel J. Fresen"
  ],
  "comment": "63 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We use and modify the Gaussian concentration inequality to prove a variety of concentration inequalities for a wide class of functions and measures on $\\mathbb{R}^{n}$, typically involving independence, various types of decay (including exponential, Weibull $0<q<1$, and polynomial) and the distribution of the gradient. We typically achieve sub-Gaussian bounds that match the variance up to a parameter independent of $n$. These methods also yield significantly shorter and simpler proofs (sometimes with improvements) of known results, such as the distribution of linear combinations of Weibull variables, Gaussian concentration of the $\\ell_{p}^{n}$ norm, and the distribution of the $\\ell_{p}^{n}$ norm on the $\\ell_{q}^{n}$ ball. We prove sub-Gaussian bounds for linear combinations of random variables with finite $r$-moments ($r>2$) that go deeper into the tails than does the weighted Berry-Esseen inequality. We use these methods to study random sections of convex bodies, proving a variation of Milman's general Dvoretzky theorem for non-Gaussian random matricies with i.i.d. entries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-28T09:52:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "60E15",
      "46N30",
      "52A23"
    ],
    "keywords": [
      "gaussian concentration inequality",
      "linear combinations",
      "milmans general dvoretzky theorem",
      "extensions",
      "typically achieve sub-gaussian bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}